On modified Newton methods with cubic convergence Jisheng Kou a, * , Yitian Li a , Xiuhua Wang b a State Key Laboratory of Water Resources and Hydropower Engineering Sciences, Wuhan University, Wuhan 430072, China b School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China Abstract In this paper, we present two new modified Newton methods for solving a non-linear equation, permitting f 0 (x) = 0 in some points. These new methods are showed to be cubically convergent. From the practical application, we show that these new methods are vast superior and possible in global convergence. Ó 2005 Elsevier Inc. All rights reserved. Keywords: Newton method; Third-order convergence; Non-linear equations; Root-finding; Iterative method 1. Introduction Newton method for a single non-linear equation (or a system of non-linear equations) is an important and basic method [1], which converges quadratically. Recently, a family of new modifications of Newton method with cubic convergence has been developed in [2–7]. In this family of new methods, there are two new methods with the best efficiency, which will be recovered in Section 2. These methods are very important and interesting because they do not require the second derivative of f(x). We know that Newton method has a problem that the condition f 0 (x) 5 0 in a neighborhood of root x * is severe indeed for its convergence and its applications is restricted [8]. Similar to Newton method, these modified Newton methods have also such problem which restricts their applications. The similar problem is also presented by Homeier in [5]. For resolving such problem of Newton method, in [8], Wu proposes a family of new continuation Newton- like methods for finding a zero of a univariate function, in which f 0 (x) = 0 is permitted in some points. In [9], we extend this result to the multi-dimensional case and present an efficient Newton-like method for vector functions, which permits the fact that the Jacobian is numerically singular in some points. But these works only concern iterative methods with quadratic convergence. Such methods with cubic convergence have not been studied and such problem of the above modified Newton methods has not been resolved. Here, by considering the discretization of HalleyÕs method [10], we recover the modifications of Newton method obtained in [2,3]. Then by considering the discretization of a variant of HalleyÕs method, we obtain two new modified Newton methods with order of convergence three, in which f 0 (x) = 0 is permitted in some 0096-3003/$ - see front matter Ó 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2005.09.052 * Corresponding author. E-mail address: koujisheng@yahoo.com.cn (J. Kou). Applied Mathematics and Computation 176 (2006) 123–127 www.elsevier.com/locate/amc